İstanbul Üniv. Fen. Fak. Mat. Der. 60 (2001), 19-32

Bornological spaces of (entire functions represented by Dirichlet series having slow growth

Mushtaq Shaker* and G.S. Srlvastava**

ABSTRACT

The study of spaces of entire functions was initiated by V.G. Iyer [6]

and the space of entire functions represented by Dirichlet series has been

studied by Hussein and Kamthan [4] and others. Patwardhan [9] has

successfully studied bornological properties of the spaces of entire function in

terms of the coefficients of Taylor series expansions. In this paper we have

used another norm and study the bornological aspects of the space F of all

00

entire Dirichlet series a{s) ~ ^ an exp(sAn) of order zero. «=1 1. INTRODUCTION

Let C denote the field of complex numbers equipped with usual

topology. Let F denote the family of all transformations:

a : C -> C such that

(1.1) a(s) = YJanexp(sAn),

19 Mushtaq Shaker arid G.S. Srivastava

where X +i > X , X] > 0 , lim X - oo , s = cr + z7 (oy real vari ables), and

;i->°o |a„}™ is any sequence of complex numbers. Set

(1.2) hmsup = /> , II -> of

Let y and T be the abscissa of convergence and abscissa of absolute convergence of a (s).

Then Bernstein [l,p.4], proved that

(1.3) 0

log la J 1 (1.4) • y ~ limsup —1—A—.

n -> «9 Xn

Thus if D* < oo and y = ao,a(s) represents an entire function and by (1.3),

T ~ oo so that the series (1.1) converges absolutely at every point of the finite complex plane. Further,, for D" = 0, we get log la I ' (1.5) x ~y -• limsup ——. n ~> 00 X

It is well known that the function a(s) is an analytic function in the half plane

a < v (--oo < T'< oo), We now assume that D* = 0 and

(1.6) r — limsup—-~—. n~>oo /I

20 Bomological spaces of entire functions represented by

It is well known that the transformation given by (1.1) represents an entire function for r = oo . Kamthan and Gautam ( [7] and [8] ) gave the set F the topology of uniform, convergence, They studied the properties of bases

of the space F using the growth properties of the entire Dirichlet series. For a e F , we set

M (a, a) s M (

Let p be a non-zero positive real number an d F now denote the family of all

entire Dirichlet functions a having Ritt order p , [11], Then every a e F can

be characterized by the condition

(1.7)

It is obvious that the above class of entire functions leaves a big subclass i.e.

those entire functions for which p = 0. To further study the growth of such

entire functions, the notion of logarithmic order is used [5], Thus an entire

function a(s) is said to be of logarithmic order p if

log log M(cr) lim sup = p, 1 < p < oo, logo-

2. DEFINITIONS

The bornological aspect for entire function have been studied by

Patwardhan [9] and others. The authors studied these properties for spaces of

entire functions represented by Dirichlet series. So far in the study of these

21 Mushtaq Shaker and G.S. Srivastava

growt h properties of entire functions have not been taken into consideration,

The {-/resent paper is an effort in this direction.

In this section we give some definitions. We have

2.1„ A homology on a set X is a family B of subsets of X satisfying the fo llowing axioms:

(\) B is a covering ofX, i.e. X = [JB;

(ii) B is hereditary under inclusion, i.e. if A e B and B is a subset of X

contained in A, then B e B ;

(iii) B is stable under finite union.

A pair (X, 'B) consisting of a set X and a homology B on X is called a bornological space, and the elements of B are called the bounded subsets of X

2.2. A base of a homology B on X is any subfamily BD of B such that every

element of Q is contained in an element of B0. A family B„ of subsets of X

is a base for a homology on X if and only if Bc covers X and every finite

union of ele ments of BD is contained in a member of B0 . Then the collection

of those subsets of X, which are contained in an element of B,( defines a

homology B on X having B0 as a base. A homology is said to be a homology with a countable base if it possesses a base consisting of a sequence of bounded sets. Such a sequence can always be assumed to be increasing.

2.3. Let E be a vector space over the complex field C. A homology Bon £ is said to be a vector homology on E, if B is stable under vector addition, homothetic transformations and the formation of circled hulls or, in other

22 Bomological spaces of entire functions represented by ...

words, if tb.e sets A+B, X A, (JF/ A belong to B, whenever A and B belong to

B and Xe C. Any pair ( E,B) consisting of vector space Fi and a vector homology on E is called a bomological vector space,

2.4. A vector homology on a vector space E is called a convex vector homology if it is stable under the formation of convex hulls. Such a homology is also stable under the formation of disked hulks, since the convex hull of a circled set is circled. A bomological vector space (E, B) whose homology B is convex is called a convex bomological vector space.

2.5. A separated bomological vector space (E, B) or (a separated homology

B) is one where {0} is the only bounded vector subspace of E .

2.6. A set P is said to be bornivorous if for every bounded set B there exists a t e C teO such that pi B c P for all fj, e C for which | ju \ < \ t\.

2.7. Let E be a vector space and let A be a disk in E not necessarily

absorbent in E. We denote by EA the vector space spanned by A, i.e. the space [JX A - [jx A. X>o A

2.8. Let E be a bomological vector space. A sequence {xn} in E is said to be

M-conver/gent to a point x e E if there exists a decreasing sequence [tn} of positive real number tending to zero such that the sequence

23 Mushtaq Shaker and G.S. Srivastava

2.9, Let E be a separated convex bomological space. A sequence \xn} in E is said to be a bomological Cauchy sequence (or a Mackey-Cauchy sequence) in

E if there exists a bounded disk B c E such that {xn} is a Cauchy sequence

in EB, For more details we refer to [3].

3. THE SPACE T

Let p > 1 be any positive real number. Further we assume that F denotes the space of all entire Dirichlet series satisfying (1.1) to (1.7) and

/-j n t' loglogM(cr) . (3.1) hmsup --£—2.—^-i- < p < +oo.

cr->« logcr

It is known [10] that (3.1) is satisfied if and only if (3.2) timsup < p -1. m <*-> loglogja,(| '"

For an entire function a (s), define the number |j a || by

iU (3.3) I a j| ~ l.u,b\\ an || ", n > 1.

For each a eF, we define

s (3.4) \\a :p + S | = £ | an\e-^ 'p^ ,

1 where S > 0 is arbitrary; On account of (3.2), (3.4) is clearly well defined. Let

f (p,5) denote the space r equipped with the norm ||.: p + 81|. We define a

homology on Twith the help of defined by (3.3). We denote by Bk the

24 Bomological spaces of entire functions represented by

set {a e f : | a || < k). Then the family B0 - IB,, ;k - 1,2,..,.} forms a base for a homology B on F.

We now prove

Theorem 3.1. (F, B) is a separated convex bomological vector space with a countable base.

Proof. Since the vector homology B on the vector space F is stable under the formation of the convex hulls, it is a convex vector homology. Since the convex hull of a circled set is circled. B is stable under the formation of disked hulls, and hence the bomological vector space (/"', B) is a convex bomological vector space. Now to show that {0} is the only bounded vector subspaces of F, we must show that F contains no bounded open set. Let

(J (s) denote the set of all a e F such that || a |j < s.

To prove the result stated, it is enough to show that no \j (s) is bounded, that in, given I) (e), we have to prove that there exists M {q) for which there is

no c > 0 such that [J (s) ce (J (if). For this purpose, take r/ ~ t Given a

positive number c , we can find a sufficiently large positive integer

A,., so that c < 2, Let exp(sA„(). Then 2)

so that c ]a

25 Mushtaq Shaker and G.S. Srivastava

does not belong to (J(?7) that is, a £ c [J(?/). This shows that {J(s) is not bounded. Thus {0} is the only bounded vector subspace of F, and hence

(F,B) is separated. Since B possesses a base consisting of increasing sequence of bounded sets, B is a homology with a countable base. Thus (F,B) is a separated convex bomological vector space with countable base. This proves

Theorem 3.1.

Theorem 3.2. B contains no bomivorous set.

Proof. Suppose B contains a bomivorous set A . Then there exists a set B. e

B such that A c. B{ and consequently/?,, is also bomivorous. We now assert

that if z, > i, then i B, <£ Bj for any / e C which leads to a contradiction. If

z, > i, it is easy to see that t Bt <£ Bi for any if. C such that | / j > 1. Now

we prove that t B,t <£ Bi for any t e C such that | / j < 1 also. Let thus

\t\ < 1. Since z, /z > 1, we can choose n such that

l 1 < 1/111 < (z; -Now let an e C be such that i° l\ t j < I a I < if" and let a = a„esX", Then \a \ = I a„ \Ui" < z, and hence a e Bi . Now \\ta\\ ta„esX" ta„ > i and hence

(a i Bi. Thus tBh <£ 5(. for any t e C. This proves Theorem 3.2

The following result is due to II. Hogbe and Nlend [2],

26 Bornotogical spaces of entire functions represented by

Theorem 3.3. The Mackey-convergence in a bornological vector space E is topologisable if and only if E has a bounded bornivorous set.

Combining Theorems 3.2 and 3.3, we get the following:

Corollary 3.1. The Mackey -convergence of F is not topologisable.

Proof. Suppose the Mackey-convergence of F is topologisable. Then by

Theorem 3.3 F has a bounded bornivorous set and this contradicts Theorem

3.2.

4. 5-NORMS ON F

We define, for each 8 > 0 , the expression

(4.1) a : p + 8II = y a\e a eF.

It is easily seen that, for each ( p,8 ), (4.1) defines a norm on the class of

entire functions represented by Dirichlet series. We shall denote by F( p,S)

the space F endowed with this norm. We denote by B(ithe homology on F

consisting of the sets bounded in the sense of the norm | a : p + 81|. We now

prove

Theorem 4.1. B = [JB(S.

Proof. Let B e B . Then there exists a constant J such that a < J for all

a„es" e B. Then a < J for all n .

27 Mushtaq Shaker and G.S. Srivastava

Choose S > 0 , such that e-(p+*Mp+*-i) < _i , J

1 Then J " < e^s)i(P+s-i) ^ or

| an | e-^P^)'(P+sH) < yA, ^(^/(^-D ( ^ > £ wliere j > o, we have

p « x \- -— - -^M

n=l ,1=1

< 00 ,

Hence B eBs and so B e (JB^. S>a

For the reverse inclusion let B eHs. then there exists a constant J such that, for all a e B, || a : p + S |j < J , i.e. f>

i.e. |«„P < j'^(e'UP^)W)j>^)/;

A i.e. || a I < /.« A. (/" " e<"^Mp+i-i j' < m _

Thus f; sB and hence [JB(S a B. This completes proof of Theorem 4,1. 8>o

28 Bornoiogical spaces of entire functions represented by

Lemma 4.1. In the topological dual F' of every functional is of the form

co -co

a a eSl anQl / (a) = cn «„ - ~ ^ n " ' i^" only if the sequence

-/!,.< p+(S)/(p c„ \e t"<5~~')} is bounded.

Proof. Suppose that / (a) is a continuous linear functional on F. Then

there exists k > 0 such that

(/ (a) j < k I a : p + (5 |j for every « .

sA Let ¥n = e « and f(yn) = cn (»>1).

In a =• ^ a,, e's,t" = Hm^fi,^ • Since / is continuous, we have

n=\ n~*° i

( "

sX f (a) = f lim V ai e " Beasts) V w « 'iimZa.'/^,) = lim > a,c

X"1

Also I c„ | < /c j ^„ : p + 8 \ = k eK (-°+s^+s~» ,

1 Then {\c Je-^-^'O^-- )| = ^e-2-t„(P+tf)/(^+i-i)

29 Mushtaq Shaker and G.S. Srivastava

A lp m0+s l) Hence {| cn \ e- " + ' } is hounded for all n = 1,2,... .

Conversely let {| c„ \ e~^s)i(P^A) j bg bounded for ali „ =1,2,....

Let / be defined by

00 00 f(a) - ~Y^cn an' a ~ Xa» e*X" ' men /'s linear functional and

n=l «=»1

00 |/(«)hZklkl ,1=1

XAp+S)l(p s ]) < jy,e- * ~ \ a„ I for some k> 0

= /c j) a : p + 51 for all a .

Hence f{d) is continuous on F. This proves Lemma 4.1.

Theorem 4.2. The bornologicai dual 7"" of F is the same as its topological dual r .

Proof. Let V be the vector space of all linear functional on the vector space /'. For every f e T and a e F we denote by < a,f > the scalar /

(a ), i.e. the value of linear functional / at the function a , the map

P x F -> K defined by (a,f) —>< a,f > is a bilinear form on Fx F called the canonical bilinear form. Let. F be the topological dual of F, i.e. the vector space of all continuous linear functionl on F. Since F' is a subspace of F , the restriction of the canonical bilinear form induces a duality between

30 Bomological spaces of entire functions represented by ...

F and F', Since F is locally convex space and separated, it follows from

Corollary 2 to Theorem 1 of [3] that this duality is separated in F.

Let the vector space F* be the set of all bounded linear functionals in

the sense of homology on F which is a separated by Theorem 3.1. We can

induce a duality between F and F * by using the bilinear form

(a,/*) —»< a,f* > = f*(a), for a e F and /* e T ', This duality is called

the bomological duality between F and F".

Let tr be the space F, endowed with the locally convex topology

associated with the homology of F . Since, algebraically, F * = (tr)' [the

topological dual of F], we see that the bomological duality between F and

F , is ideritical to the topological duality between tr and (tr)'.

Thus the proof of Theorem 4.2 is complete.

REFERENCES

[1] Bernstein, V. - Lecons sur les progres recents de la theorie des series de Dirichlet. Gauthiers-Villars, Paris, (1933).

[2] Hogbe-Nlend, H. - The'oriedes Bomologies et Applications. Lecture Notes, Springer-verlag, Berlin, No. 213, (1971).

[3] Hogbe -Nlend, H. - Bomologies and functional analysis. North- Holland publishing company, Netherlands, (1977).

[4] Hussain, Taqdir and Kamthan, P.K, - Spaces of entire functions represented by Dirichlet series, Collectania Mathematica, 19 (3), (1968): 203-216.

31 Mushtaq Shaker and G.S. Srivastava

[5] Iyer, V.G. - A property of the maximum modulus of integral functions. J. Indian Math. Soc, 6 (1942): 69-80.

[6] Iyer, V.G. - On the space o:f integral functions (I). J. Indian Math. Soc. (New Series), 12, (1948): 13-30.

[7] Kamthan, P.K. and S.K, Singh Gautam - Bases in the space of analytic Dirichlet transformations. Collect. Math. 23, (1972): 9-16.

[8] Kamthan, P.K. and S.K. Singh Gautam - Bases in a certain space of Dirichlet entire transformations. Indian J. Pure Appl. Math. 6 (8), (1975): 856-863.

[9] Patwardhan, M.D. - Study on bomological properties'of the space of integral functions. Indian J. Pure Appl. Math. 12 (7), (1981): 865-873.

[10] Prem Singh - Some aspects of the growth of entire functions and Dirichlet series, Thesis. I..I.T. Kanpur, 1966.

[11] Ritt, J.F. - A note on the proximate order of entire functions represented by Dirichlet series, Bull, de L'Academie Polonaise des Sciences, 19 (3), (1971): 199-202,

* Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India.

** Department of Mathematics, Indian Institute of Technolog)) ~ Roorkee, Roorkee 247667, India. E-mail address: [email protected]

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